Problem 8.2
library(FrF2)
## Warning: package 'FrF2' was built under R version 4.0.5
## Loading required package: DoE.base
## Warning: package 'DoE.base' was built under R version 4.0.5
## Loading required package: grid
## Loading required package: conf.design
## Warning: package 'conf.design' was built under R version 4.0.3
## Registered S3 method overwritten by 'DoE.base':
## method from
## factorize.factor conf.design
##
## Attaching package: 'DoE.base'
## The following objects are masked from 'package:stats':
##
## aov, lm
## The following object is masked from 'package:graphics':
##
## plot.design
## The following object is masked from 'package:base':
##
## lengths
des.res5<-FrF2(nfactors=4,resolution=5,randomize=TRUE)
## full factorial design needed
## creating full factorial with 16 runs ...
aliasprint(des.res5)
## NULL
response <- c(7.037,14.707,11.635,17.273,10.403,4.368,9.360,13.440,8.561,16.867,13.876,19.824,11.846,6.125,11.190,15.653)
des.resp<-add.response(des.res5,response)
summary(des.resp)
## Call:
## fac.design(2, nfactors, factor.names = factor.names, replications = replications,
## repeat.only = repeat.only, randomize = randomize, seed = seed)
##
## Experimental design of type full factorial
## 16 runs
##
## Factor settings (scale ends):
## A B C D
## 1 -1 -1 -1 -1
## 2 1 1 1 1
##
## Responses:
## [1] response
##
## The design itself:
## A B C D response
## 1 1 -1 -1 -1 7.037
## 2 -1 1 -1 1 14.707
## 3 1 1 -1 -1 11.635
## 4 1 -1 -1 1 17.273
## 5 1 1 1 -1 10.403
## 6 -1 1 1 1 4.368
## 7 -1 1 1 -1 9.360
## 8 -1 1 -1 -1 13.440
## 9 -1 -1 1 -1 8.561
## 10 1 -1 1 1 16.867
## 11 1 1 -1 1 13.876
## 12 -1 -1 1 1 19.824
## 13 1 -1 1 -1 11.846
## 14 -1 -1 -1 -1 6.125
## 15 -1 -1 -1 1 11.190
## 16 1 1 1 1 15.653
## class=design, type= full factorial
Problem 8.24
A <- rep(c(-1,1),8)
B <- rep(c(-1,-1,1,1),4)
C <- rep(c(rep(-1,4),rep(1,4)),2)
D <- c(rep(-1,8),rep(1,8))
E <- A*B*C*D
effects <- c("e","a","b","abe","c","ace","bce","abc","d","ade","bde","abd","cde","acd","bcd","abcde")
AB <- A*B
Block <- c(1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1)
des <- as.data.frame(cbind(A,B,C,D,E,effects,AB,Block))
colnames(des) <- c("A","B","C","D","E=ABCD","observation","Blocks=AB","Block")
des
## A B C D E=ABCD observation Blocks=AB Block
## 1 -1 -1 -1 -1 1 e 1 1
## 2 1 -1 -1 -1 -1 a -1 2
## 3 -1 1 -1 -1 -1 b -1 2
## 4 1 1 -1 -1 1 abe 1 1
## 5 -1 -1 1 -1 -1 c 1 1
## 6 1 -1 1 -1 1 ace -1 2
## 7 -1 1 1 -1 1 bce -1 2
## 8 1 1 1 -1 -1 abc 1 1
## 9 -1 -1 -1 1 -1 d 1 1
## 10 1 -1 -1 1 1 ade -1 2
## 11 -1 1 -1 1 1 bde -1 2
## 12 1 1 -1 1 -1 abd 1 1
## 13 -1 -1 1 1 1 cde 1 1
## 14 1 -1 1 1 -1 acd -1 2
## 15 -1 1 1 1 -1 bcd -1 2
## 16 1 1 1 1 1 abcde 1 1
One two factor interaction and one three factor interaction are confounded with the blocks. These interactions are AB and CDE.
Problem 8.25
A <- rep(c(-1,1),16)
B <- rep(c(-1,-1,1,1),8)
C <- rep(c(rep(-1,4),rep(1,4)),4)
D <- rep(c(rep(-1,8),rep(1,8)),2)
E <- c(rep(-1,16),rep(1,16))
F_factor <- C*D*E
G <- A*B*C
effects <- c("(1)","ag","bg","ab","cfg","acf","bcf","abcfg","df","adfg","bdfg","abdf","cdg","acd","bcd","abcdg","ef","aefg","befg","abef","ceg","ace","bce","abceg","de","adeg","bdeg","abde","cdefg","acdef","bcdef","abcdefg")
ACE <- A*C*E
BFG <- B*F_factor*G
Block <- c(1,4,1,4,3,2,3,2,2,3,2,3,4,1,4,1,4,1,4,1,2,3,2,3,3,2,3,2,1,4,1,4)
des <- as.data.frame(cbind(A,B,C,D,E,F_factor,G,effects,ACE,BFG,Block))
colnames(des) <- c("A","B","C","D","E","F=CDE","G=ABC","observation","Blocks=ACE","Block=BFG","Block")
des
## A B C D E F=CDE G=ABC observation Blocks=ACE Block=BFG Block
## 1 -1 -1 -1 -1 -1 -1 -1 (1) -1 -1 1
## 2 1 -1 -1 -1 -1 -1 1 ag 1 1 4
## 3 -1 1 -1 -1 -1 -1 1 bg -1 -1 1
## 4 1 1 -1 -1 -1 -1 -1 ab 1 1 4
## 5 -1 -1 1 -1 -1 1 1 cfg 1 -1 3
## 6 1 -1 1 -1 -1 1 -1 acf -1 1 2
## 7 -1 1 1 -1 -1 1 -1 bcf 1 -1 3
## 8 1 1 1 -1 -1 1 1 abcfg -1 1 2
## 9 -1 -1 -1 1 -1 1 -1 df -1 1 2
## 10 1 -1 -1 1 -1 1 1 adfg 1 -1 3
## 11 -1 1 -1 1 -1 1 1 bdfg -1 1 2
## 12 1 1 -1 1 -1 1 -1 abdf 1 -1 3
## 13 -1 -1 1 1 -1 -1 1 cdg 1 1 4
## 14 1 -1 1 1 -1 -1 -1 acd -1 -1 1
## 15 -1 1 1 1 -1 -1 -1 bcd 1 1 4
## 16 1 1 1 1 -1 -1 1 abcdg -1 -1 1
## 17 -1 -1 -1 -1 1 1 -1 ef 1 1 4
## 18 1 -1 -1 -1 1 1 1 aefg -1 -1 1
## 19 -1 1 -1 -1 1 1 1 befg 1 1 4
## 20 1 1 -1 -1 1 1 -1 abef -1 -1 1
## 21 -1 -1 1 -1 1 -1 1 ceg -1 1 2
## 22 1 -1 1 -1 1 -1 -1 ace 1 -1 3
## 23 -1 1 1 -1 1 -1 -1 bce -1 1 2
## 24 1 1 1 -1 1 -1 1 abceg 1 -1 3
## 25 -1 -1 -1 1 1 -1 -1 de 1 -1 3
## 26 1 -1 -1 1 1 -1 1 adeg -1 1 2
## 27 -1 1 -1 1 1 -1 1 bdeg 1 -1 3
## 28 1 1 -1 1 1 -1 -1 abde -1 1 2
## 29 -1 -1 1 1 1 1 1 cdefg -1 -1 1
## 30 1 -1 1 1 1 1 -1 acdef 1 1 4
## 31 -1 1 1 1 1 1 -1 bcdef -1 -1 1
## 32 1 1 1 1 1 1 1 abcdefg 1 1 4
In this design, the blocks are confounded with ACE, BFG, and ABCEFG. So, no two factor interactions were confounded with the blocks.
Problem 8.28
I <- c(0.0167,0.0062,0.0041,0.0073,0.0047,0.0219,0.0121,0.0255,0.0032,0.0078,0.0043,0.0186,0.0110,0.0065,0.0155,0.0093)
II <- c(0.0128,0.0066,0.0043,0.0081,0.0047,0.0258,0.0090,0.00250,0.0023,0.0158,0.0027,0.0137,0.0086,0.0109,0.0158,0.0124)
III <- c(0.0149,0.0044,0.0042,0.0039,0.0040,0.0147,0.0092,0.0226,0.0077,0.0060,0.0028,0.0158,0.0101,0.0126,0.0145,0.0110)
IV <- c(0.0185,0.0020,0.0050,0.0030,0.0089,0.0296,0.0086,0.0169,0.0069,0.0045,0.0028,0.0159,0.0158,0.0071,0.0145,0.0133)
des.res4<-FrF2(nfactors=6,resolution=4,randomize=FALSE)
des.resp<-add.response(des.res4,IV)
summary(des.resp)
## Call:
## FrF2(nfactors = 6, resolution = 4, randomize = FALSE)
##
## Experimental design of type FrF2
## 16 runs
##
## Factor settings (scale ends):
## A B C D E F
## 1 -1 -1 -1 -1 -1 -1
## 2 1 1 1 1 1 1
##
## Responses:
## [1] IV
##
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D E=E F=F
##
## $generators
## [1] E=ABC F=ABD
##
##
## Alias structure:
## $fi2
## [1] AB=CE=DF AC=BE AD=BF AE=BC AF=BD CD=EF CF=DE
##
##
## The design itself:
## A B C D E F IV
## 1 -1 -1 -1 -1 -1 -1 0.0185
## 2 1 -1 -1 -1 1 1 0.0020
## 3 -1 1 -1 -1 1 1 0.0050
## 4 1 1 -1 -1 -1 -1 0.0030
## 5 -1 -1 1 -1 1 -1 0.0089
## 6 1 -1 1 -1 -1 1 0.0296
## 7 -1 1 1 -1 -1 1 0.0086
## 8 1 1 1 -1 1 -1 0.0169
## 9 -1 -1 -1 1 -1 1 0.0069
## 10 1 -1 -1 1 1 -1 0.0045
## 11 -1 1 -1 1 1 -1 0.0028
## 12 1 1 -1 1 -1 1 0.0159
## 13 -1 -1 1 1 1 1 0.0158
## 14 1 -1 1 1 -1 -1 0.0071
## 15 -1 1 1 1 -1 -1 0.0145
## 16 1 1 1 1 1 1 0.0133
## class=design, type= FrF2
DanielPlot(des.resp,half=TRUE)
MEPlot(des.resp,show.alias=TRUE)
(a) \(2_{IV}^6-2\) , a 16-run design
(b) The defining relation is I = ABCE = ACDF = BDEF, the alias structure can be found in Table 8.8.
(c) From the half normal plot, variables A, C, and D affect the average number.
(d)
(e)
Problem 8.40
(a) 4 factors
(b) Resolution = III
(c)
(d)
Problem 8.48
(a) The generator is interaction AB
(b) The generator is interaction AC
(c) Resolution = III
Problem 8.60
The alias relationships are ABD, ACE, BCF, and ABCG
R Code
library(FrF2)
des.res5<-FrF2(nfactors=4,resolution=5,randomize=TRUE)
aliasprint(des.res5)
response <- c(7.037,14.707,11.635,17.273,10.403,4.368,9.360,13.440,8.561,16.867,13.876,19.824,11.846,6.125,11.190,15.653)
des.resp<-add.response(des.res5,response)
summary(des.resp)
A <- rep(c(-1,1),8)
B <- rep(c(-1,-1,1,1),4)
C <- rep(c(rep(-1,4),rep(1,4)),2)
D <- c(rep(-1,8),rep(1,8))
E <- A*B*C*D
effects <- c("e","a","b","abe","c","ace","bce","abc","d","ade","bde","abd","cde","acd","bcd","abcde")
AB <- A*B
Block <- c(1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1)
des <- as.data.frame(cbind(A,B,C,D,E,effects,AB,Block))
colnames(des) <- c("A","B","C","D","E=ABCD","observation","Blocks=AB","Block")
des
A <- rep(c(-1,1),16)
B <- rep(c(-1,-1,1,1),8)
C <- rep(c(rep(-1,4),rep(1,4)),4)
D <- rep(c(rep(-1,8),rep(1,8)),2)
E <- c(rep(-1,16),rep(1,16))
F_factor <- C*D*E
G <- A*B*C
effects <- c("(1)","ag","bg","ab","cfg","acf","bcf","abcfg","df","adfg","bdfg","abdf","cdg","acd","bcd","abcdg","ef","aefg","befg","abef","ceg","ace","bce","abceg","de","adeg","bdeg","abde","cdefg","acdef","bcdef","abcdefg")
ACE <- A*C*E
BFG <- B*F_factor*G
Block <- c(1,4,1,4,3,2,3,2,2,3,2,3,4,1,4,1,4,1,4,1,2,3,2,3,3,2,3,2,1,4,1,4)
des <- as.data.frame(cbind(A,B,C,D,E,F_factor,G,effects,ACE,BFG,Block))
colnames(des) <- c("A","B","C","D","E","F=CDE","G=ABC","observation","Blocks=ACE","Block=BFG","Block")
des
I <- c(0.0167,0.0062,0.0041,0.0073,0.0047,0.0219,0.0121,0.0255,0.0032,0.0078,0.0043,0.0186,0.0110,0.0065,0.0155,0.0093)
II <- c(0.0128,0.0066,0.0043,0.0081,0.0047,0.0258,0.0090,0.00250,0.0023,0.0158,0.0027,0.0137,0.0086,0.0109,0.0158,0.0124)
III <- c(0.0149,0.0044,0.0042,0.0039,0.0040,0.0147,0.0092,0.0226,0.0077,0.0060,0.0028,0.0158,0.0101,0.0126,0.0145,0.0110)
IV <- c(0.0185,0.0020,0.0050,0.0030,0.0089,0.0296,0.0086,0.0169,0.0069,0.0045,0.0028,0.0159,0.0158,0.0071,0.0145,0.0133)
des.res4<-FrF2(nfactors=6,resolution=4,randomize=FALSE)
des.resp<-add.response(des.res4,IV)
summary(des.resp)
DanielPlot(des.resp,half=TRUE)
MEPlot(des.resp,show.alias=TRUE)